A general description of criticality in neural network models

Abstract

Recent experimental observations have supported the hypothesis that the cerebral cortex operates in a dynamical regime near criticality, where the neuronal network exhibits a mixture of ordered and disordered patterns. However, A comprehensive study of how criticality emerges and how to reproduce it is still lacking. In this study, we investigate coupled networks with conductance-based neurons and illustrate the co-existence of different spiking patterns, including asynchronous irregular (AI) firing and synchronous regular (SR) state, along with a scale-invariant neuronal avalanche phenomenon (criticality). We show that fast-acting synaptic coupling can evoke neuronal avalanches in the mean-dominated regime but has little effect in the fluctuation-dominated regime. In a narrow region of parameter space, the network exhibits avalanche dynamics with power-law avalanche size and duration distributions. We conclude that three stages which may be responsible for reproducing the synchronized bursting: mean-dominated subthreshold dynamics, fast-initiating a spike event, and time-delayed inhibitory cancellation. Remarkably, we illustrate the mechanisms underlying critical avalanches in the presence of noise, which can be explained as a stochastic crossing state around the Hopf bifurcation under the mean-dominated regime. Moreover, we apply the ensemble Kalman filter to determine and track effective connections for the neuronal network. The method is validated on noisy synthetic BOLD signals and could exactly reproduce the corresponding critical network activity. Our results provide a special perspective to understand and model the criticality, which can be useful for large-scale modeling and computation of brain dynamics.

Keywords: Bifurcation; Criticality; Ensemble Kalman filter; Neuronal avalanches.

Algorithm 1

EnKF.

Figure 1

Spiking activity with respect to the excitatory conductance (A): Network activity (firing rate, coefficient of variance and coherence coefficient with respect to different AMPA conductance while keeping the constraint $\mathrm{AMPA}+\mathrm{20}\times \mathrm{NMDA}=\mathrm{2.5}$. There exists an obvious sharp transition from an asynchronous to a synchronous state in the one-dimensional transition, in which the boundary is characterized as a moderately synchronized state, as shown in the middle panel of (B). (B): The local recurrent neuronal network consists of Exc. and Inh. spiking neurons with different AMPA levels. Network activity of three typical points with parameters indicated in the top panel. middle, raster plot of a subset of 200 neurons (Exc. 160 (black), Inh. 40 (red)); bottom, the average excitatory and inhibitory synaptic currents.

Figure 2

phase diagram and spiking avalanches of the neuronal network. (A, B, C): The phase diagram of the network with respect to Δ and AMPA conductance in the mean-dominated driven case. The parameter space suggests a phase transition, as shown in the heatmap of (C) the κ coefficient and (E) the power of collective oscillations. (D, E, F): the same illustration as the top panel but in the condition of fluctuation-dominated current.

Figure 3

Inhibition modulates the oscillation frequency. (A) Oscillations in the model of criticality and PSD of the synaptic current for the inhibitory time constant τ_i = 10 ms. (B) The different oscillations with much smaller frequencies are shown on the right holding τ_i = 12 ms for all neurons.

Figure 4

Critical state with spike avalanches. (A): mapping spikes from N_s randomly sampled excitatory neurons into time bins (Δt = 0.5 ms). Here an avalanche event is defined as a sequence of time bins in which the spiking count at least exceeds Θ, ending with a “silent” time bin. (B, C): Typical distributions of avalanche size and avalanche duration for networks with different synaptic parameters in different regions. At different levels of AMPA conductance, the model may present subcritical (blue), critical (black), and supercritical (red) avalanche dynamics. Inset: at the critical state, the average size S conditioned on a given duration T shows power-law increases corresponding to S ∼ T^π. (D, E): The corresponding distributions of avalanche events for networks in fluctuation-dominated regime. Similarly, three levels of synaptic conductance are considered and the model only exhibits subcritical dynamics due to high fluctuation among neurons.

Figure 5

Critical exponents vary in different recording sizes. (A): Avalanche assessment in different sampling sizes. (B) Corresponding distributions of avalanche duration at different sampling sizes.

Figure 6

Dynamics of the field model (A): The evolution of their gating variable and network firing rate in the parameter of power-law criticality. (B): The trajectories of S_s and S_i in the supercritical parameter and represented as a limit cycle.

Figure 7

Tracking the critical and subcritical dynamics in the network. (A): The assimilation process of the network in tracking critical dynamics. The filtered signal is almost consistent with the ground truth (top panel). The filtered AMPA weights are plotted in the assimilation process (bottom panel). The dashed blue region represents the deviation among ensemble members. Note that the parameters converge rapidly to the ground truth of the critical parameters. (B): The same as (A) but in the subcritical state. (C) The ground truth and the resimulated BOLD signal show a correlation of 0.33 after a period of transition. The simulated signal tracks the critical BOLD signal in real-time. (D) The simulated network reaches a correlation coefficient of 0.60 with its biological counterpart at the subcritical state.